RAIRO. R ECHERCHE OPÉRATIONNELLE
S. S OFIANOPOULOU
A queueing network application to a telecommunications distributed system
RAIRO. Recherche opérationnelle, tome 26, no4 (1992), p. 409-420
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Recherche opérationnelle/Opérations Research (vol 26, n° 4, 1992, p. 409 à 420)
A QUEUEING NETWORK APPLICATION TO A TELECOMMUNICATIONS DISTRIBUTED SYSTEM (*)
by S. SOFIANOPOULOU O
Communicated by R. E. BURKARD
Abstract. — The purpose of this paper is to present andsolve a particular class of a télécommuni- cations related Process Allocation Problem. The problem deals with the allocation of processes to a network of processors with the aim to minimize a "trade-off" objective function composed of(a) the queueing delays overhead which is formed in the underlying queueing network and (Jb) the communication costs incurred between processes residing on different processors. Various application constraints are also taken into account. A cost function is first constructed to reflect the queueing delay "feit" by a subscriber and a simulated annealing algorithm is then used to minimize the trade-off objective function.
Keywords : Queueing networks; distributed processing; process allocation.
Résumé. — L'objet de cet article est de présenter et résoudre une classe particulière de problème d'affectation continue de processus (entendu ici comme modules, programmes spéciaux, etc.) relatifs à un système de télécommunication. Le problème traite de l'affectation de processus un réseau de processeurs en vue de minimiser une fonction-objectif de substituabilité composée de: (a) les charges dues aux retards dans le réseau de files d'attente sous-jacent; (b) les coûts dûs aux communications entre les processus résidant dans les différents processeurs. Nous prenons aussi en compte diverses contraintes. Une fonction de coût est tout d'abord construite pour refléter le retard «ressenti» par un abonné, puis nous utilisons un algorithme de recuit simulé pour minimiser la fonction de substituabilité.
Mots clés : Réseaux de files d'attente; traitement distribué; affectation de processus.
1. INTRODUCTION
Of primary importance in modem computer and communication technol- ogy is the concept of distributed Systems. Distributed Systems are considered to be networks of loosely coupled processors on which jobs are executed. A particular job in the System is partitioned into several small self-contained tasks which are performed by different independent modules, special pro-
(*) Received May 1990.
(*) The Economie University of Athens, 104 34 Athens, Greece.
Recherche opérationnelle/Opérations Research, 0399-0559/92/04 409 12/$ 3.20
grams, known as processes. In a télécommunications environment, which is of interest hère, the processors System controlling a téléphone exchange is considered as a distributed processing network where a variety of jobs, such as fault interrupts, administrative programs, call processing, routine tests and diagnostics are performed by the processors.
As the job is progressed, every task is processed by a particular process, while several other tasks, belonging to either the same or a different job, are due to be executed by the same process. This gives rise to delays, since the chain of tasks which form a job must wait until the appropriate process becomes available and ready to exécute them. The queueing delays overhead is a significant cost in the problem which needs to be minimized in order to improve the network's performance.
Another "cost" which is also associated with the distributed network is the message passing overhead related to the exchange of messages among processes residing on different processors. The problem of allocating processes to processors with the aim to minimize the message passing overhead is known as the Process Allocation Problem (PAP) [11]. In télécommunications applications, however, one is faced with the problem of allocating processes to processors, so that both the queueing delay and communication costs are suffïciently low. In addition, certain resource requirements -constraints- should be met.
The two above mentioned performance criteria are obviously conflicting.
It is evident that in order to keep both costs at a reasonable (or désirable) level, some sort of trade-off between the queueing delay and the message passing cost, associated with a particular network configuration, should be adopted.
The purpose of this paper is to introducé the queueing delay overhead to the PAP as a performance criterion via a trade-off approach. The resulting minimization problem is tackled with an efficient simulated annealing algo- rithm. Computational results of a set of random problems which have similar characteristics to a "real-world" application in télécommunications are reported. A brief discussion of the application of queueing network theory to the PAP is also included.
2. PROBLEM STATEMENT
Consider a number N of processes not necessarily all distinct from one another. Replicate processes can be included in the problem with the restric- tion not to co-exist in the same processor. Let C dénote the message passing
QUEUING NETWORK IN TELECOMMUNICATIONS DISTRIBUTED SYSTEM 4 1 1
matrix whose éléments cip i=l, . . . , TV— 1, y = z + 1, . . ., N, represent the amount of messages exchanged between processes i and y residing on different processors (groups of processes). It is assumed that the communication cost is zero when processes / and j réside on the same processor. The sum of messages received by process i is denoted by mt, whilst rt is the resource requirement (in memory, occupancy etc.) of process i and R is the resource availability (in memory, occupancy etc.) of each one of the identical proces- sors.
The minimal combined cost (of communication and queueing delay) prob- lem is formulated as a 0-1 programming problem using the 0,1 variables XtjJ i~ 1, . . . , i V - 1, j=i+ 1, . . .,N. Xtj is 1 if processes i and j are co-located, allocated to the same processor, and 0 if they réside on different processors.
The formulation of the problem is
minimize ƒ = wfc + (1 — w)fd (1) subject to
riXik + X rjX]cjf*R~rk (memory/capacity constraints)
j = k+l
* = l , . . . , t f (1.1) (triangular constraints)
— Y 4- Y < 1 i — i AT—? n ?"i n Aik^Aik=l z — i , . . . , J V z , U-A>
1, if processes i and j are allocated to the same
processor (co-located) (1.3) 0, otherwise, i= 1, . . . ,iV— 1, 7 = / + 1, . . .,iV~
Zl7 = 0, if i and j are replicate processes (1.4) where fc and /d are the two performance criteria, i. e. the communication and queueing delays costs respectively, appropriately scaled to be rendered in the same order of magnitude. These two costs are discussed in detail in the next section.
w is a weighting factor ranging from 0 to 1 indicating the direction of
"interest" between the two costs. At the two extreme values of w, i.e. at w = 0 and w = 1, we seek to minimize either the amount of queueing delay (which is trivial, since this will produce allocations which use as many processors as there are processes) or the inter-processor communication
overhead. At intermediate values of w, i. e. as w gradually increases from 0 to 1, we obtain solutions with the weight moving from solutions with near- to-minimum amount of queueing delays to those with near-to-minimum amount of communication cost.
It is interesting to note that problem formulation (1) is a mathematical programming formulation which does not only produce solutions with the minimal combined cost, but it also provides automatically the number of groups (processors) to which the best allocation corresponds [12].
3. PERFORMANCE CRITERIA The Queueing Network Model
A queueing network can be defined as a collection of service centres, L e.
a multiple resource System, where the customers proceed from one to another in order to fùlfill their service requirements [5]. When queueing networks are related to the PAP, in any particular solution/allocation the service centres represent the processes (and hence the processors which exécute them) and the customers represent the various tasks to which the call processing is divided. Each service centre has its own queue in which customers wait to receive service. There is a certain service rate and a queueing discipline associated with each one of these centres.
In the present analysis Jackson's model is employed. Jackson [7] introduced the analysis of open queueing networks. A network is considered open if jobs are permitted to enter or leave the System at any time. He proved that in equilibrium each service centre behaves as if it were an individual M/M/1 queueing System, i. e. a System with Poisson arrivai rates, exponential service rates, FCFS queue discipline and a single server. This model has the property of having a stationary solution in "product form" for the joint probabilîties of the lengths of the queues, L e.
P(nl9n2, . . •,nQ)=p1(n1)p2(n2). . .pQ(nQ)
where pt (nt) represents the marginal probability that there are nt customers waiting or in service at centre i. This resuit, known as Jackson's décomposi- tion theorem, proves that the joint distribution is decomposable into the product of Q marginal distributions. Thus an open queueing network of service centres can be decomposed into several individuaî centres, and its performance measures can be calculated using the traditional queueing theory for M/M/1 queueing Systems (e. g. see [4]).
QUEUING NETWORK ÏN TELECOMMUNICATIONS DISTRIBUTED SYSTEM 4 1 3
Application to PAP
In calculating the queueing delays in the PAP Jackson's model has been adopted. It is assumed that the flows of messages int o and out of each processor are equal. Although this is not always true in a real system (some processes expire at certain stages and hence do not forward any more messages) it does not seriously affect the results [9].
Each service centre is considered as an independent M/M/1 queueing system with one server serving at a time. Although each service centre (processor) has several servers (processes) allocated to it, it is still considered as an M/M/1 queueing system since only one process can be executed at a time by each processor. The service rate [xt of each process i run on any processor dépends on the load of the processor on which it résides (group of co-located processes) and is given by
Vt^i/Pt (2)
where À,- is the sum of messages received by process i as well as any other process co-located to it, and pi is the sum of their occupancies, z. e. proportion of time for which they are running. Thus
i - l JV
Xi=ml+Y,mJXjt+ £ mkXik (3)
j = 1 k - i + 1 i - i AT
Pi = ri+lrjXji+ £ rkXik (4)
j=l k=i+l
Then the delay "seen" by a task requiring service by process i is given by 4 = 1 / 0 1 , - ^ (5) This delay includes both service and queueing time and can be written as
4 =P j/ M i - P i ) (6)
The overall delay of a job submitted to a network of processors consists of the sum of all delays experienced by its chain of tasks as the latter are sequentially executed by different processes allocated to the various proces- sors. Our objective here, however, is not to minimize this overall delay but only the delay which is "feit" by the network user, in our case the téléphone system user-subscriber. During the call setup procedure (job), for instance, the subscriber "notices" the delay associated with the process of analyzing
the fîrst few (routing) digits dialed, but he does not "notice" any delay involved with the traffïc-recording process.
What is of importance therefore, is the sum of the delays on certain time- critical paths of the job. These critical paths, however, are not ail equally important because some of them are used by the call setup procedure more frequently than others. Since it is most unusual to minimize the delays on ail critical paths simultaneously, what we are interested in, is to minimize a single "effective" delay composed of the weighted delays on each time-critical path, the weighting factors being proportional to the percentage of jobs, call setup procedures, using each path.
Assume for example that we have s time-critical paths denoted by Pk, fc=l, . . .,$, each composed by a séquence of a number of processes l(k) kl9k29 . . .9kq9 . . -,klik)i where kq could be any number between 1 and N.
Let dPl9 dP2, . . .9dPs be the delays on each path respectively and let ak9
k=l9 . . . ,5 be the percentage (frequency) of the call setups using time-critical path k (Lak=l), Then the "effective" delay sought to be minimized would be
. . . +akdPk+ . . . +asdPs (7) where dPk = dx
Equation (7) becomes
p i fis
/ < = X * I
X (lPK_
0)k=l q=l Af c ( 1 Pk )
or in terms of Xy
s Jd~~ ZJ ak
l {k) %
q=l
i = ka+l
x M - ' * , - E rjXJkt- X riXkqi
\ i = kq+l
The fractional type of objective function (8) is typical of queueing delays costs [2].
QUEUING NETWORK IN TELECOMMUNICATIONS DISTRIBUTED SYSTEM 4 1 5
Communication cost
The other aspect of the PAP, which is mentioned in the first section, is the message passing cost associated with the volume of messages exchanged between processes assigned to different processors. Processes that communi- cate with one another incur a communication cost cip i^j, and the objective would be to minimize the message passing cost between processes residing in different processors, i. e.
N - l N
/c=E I CyO-JTy) (9)
The sum of équations (8) and (9) form the optimization criterion of the PAP considered in this work.
4. THE ALGORITHM
In formulation (1) the communication cost part of the objective function as well as the constraints are linear in the Xtj variables. On the other hand, the queueing delays overhead part of the objective function is fractional.
Even if only the communication cost is taken into account, the PAP is more gênerai than the NP-hard graph partitioning problem. It seems therefore appropriate to tackle the PAP using a heuristic algorithm.
The heuristic algorithm employed to solve problem (1) is simulated anneal- ing [10], [14], [15]. This algorithm is a heuristic optimization method based on itérative improvement. The method has been successfully applied to combinatorial problems in computer Systems design [8], to the solution of the travelling salesman [6], [8], and the quadratic assignment problems [3], and to the minimization of message passing cost in the PAP [13].
The basic idea is to generate random displacements from any current feasible solution, and accept as new current solution not only solutions which improve the objective function, but also some, which do not improve it; the latter ones are accepted with probability exp( — Af/T), depending on the amount of détérioration Af of the objective function and a tunable parameter T (the température).
The two ingrédients required in the implementation of simulated annealing are:
1. A perturbation scheme for generating random displacements, i. e, a neighborhood génération scheme.
2. An annealing schedule. This consists of determining
(a) A séquence of the control parameter température T; i,e. initial and final values and a rate of decrease.
(b) The number of solutions Lt attempted at each température Tt as the température decreases.
The algorithm itérâtes until the stopping température value is reached or no feasible solution has been found during Lt attempts.
In the present implementation, the neighborhood génération scheme con- sists of randomly selecting two processes p and q which are not interconnected (Xpq^0)y and making the corresponding variable Xpq equal to 1.. Of course this move has some implications which involve disconnecting process p from ail processes to which it was previously connected and Connecting it to ail processes to which q is already connected [triangular constraints (1.2)].
The initial (and final) value of T is determined using a small number of pilot runs before the actual annealing process begins, at an appropriately high (low) value so that almost ail candidate solutions, which deteriorate the objective function, are accepted as current solution with a probability of 0.95 or more (0.05 or less). The cooling schedule, /. e, the rate of decrease of température T, is very crucial for the successful application of the algorithm.
If the rate of température decrease is too high then the algorithm leads to local optimum solutions, while if it is too low CPU time is wasted. In this work the cooling schedule suggested in [I] is adopted where Tt is updated by
T T<
where 5 = 0.1 and at is the standard déviation of the objective function values of the solutions examined at Tv
The number of solutions Lt attempted at each température Tt is set to N(N— l)/2. This parameter setting is adopted taking into account that all possible moves that can lead a current solution to a neighboring one (/. e.
making a variable Xpqi which is currently set to 0, equal to 1) are at most N(N- l)/2 (actually they are much fewer since capacity and triangular con- straints should be satisfied).
Having defined the above ingrédients of the simulated annealing and an initial feasible solution to the PAP, one can apply the algorithm. Obviously, the number of itérations to be performed dépends on the size of the problem.
For the sake of completeness it should be mentioned that a starting feasible solution to the PAP is formed by randomly choosing processes and grouping
QUEUING NETWORK IN TELECOMMUNICATIONS DISTRIBUTED SYSTEM 4 1 7
them together until a capacity constraint is violated. If this is the case, the next randomly chosen process starts forming a second group and this procedure is continued until all processes are grouped.
5. COMPUTATIONAL RESULTS
Computations were carried out on a VAX 8810 computer. The 24 randomly generated data sets used to test the method are very similar in structure to a 12-processes sample instance of the problem which was provided to us by a télécommunications laboratory. Data input to the model includes the number of processes to be allocated (TV), the occupancy, code- and data-storage requirements (rt) of each process i, and the resource availabilities on each processor (R), the matrix C=[c^] of messages exchanged between processes and the time-critical paths Pk, with their usage frequencies ak. The random data sets include 12 problems with 12 processes each, while replicate processes (identical to one another which are not allowed to run on the same processor) are present in the model. Different groups of the above test problems were generated, having 5 sets with three (12-3GRP1 to 12-3GRP5), 2 sets with one (12-1GRP1 and 12-1GRP2), 5 sets with zero (12-OGRP1 to 12-OGRP5) group(s) of three replicate processes each. Larger test problems, compared to our original instance of the problem, including 5 problems with 15 pro- cesses each (15-OGRP1 to 15-OGRP5), 5 problems with 20 processes each (20-OGRP1 to 20-OGRP5) and finally 2 problems, each with 25 processes (25-OGRP1 and 25-OGRP2) were also tried. For each one of the identical processors used in the model the occupancy figure did not exceed 70%, while the amount of code- and data-storage did not exceed 300 and 350 units respectively. The time-critical paths used to evaluate the queueing delays in each test problem were constructed by randomly choosing séquences of processes.
Tables 1 and 2 present the results obtained with the weight w of the combined objective function equal to 0.8 and 0.2 respectively. The two tables are divided in three parts. The annealing heuristic was run with (a) Lt = N(N- l)/2, (b) Lt = N(N~ l)/4 and (c) Lt = N(N- l)/8 and the mes- sage passing, the amount of queueing delay and the CPU seconds consumed in each case are recorded in each one of the three parts of the tables. It is reasonable to assume that the annealing schedule with Lt = N(N— l)/2 would provide solutions closer to the optimum, which is of course not known, since the algorithm spends more time in each température attempting more solutions. Comparing now the results for Lt = N(N— l)/2 and Lt = N(N~ l)/4
DATA SET 12-3GRP1 12-3GRP2 12-3GRP3 12-3GRP4 12-3GRP5 12-1GRP1 12-1GRP2 12-0GRP1 12-0GRP2 12-0GRP3 12-0GRP4 12-0GRP5 15-0GRP1 15-0GRP2 15-0GRP3 15-0GRP4 15-0GRP5 20-0GRP1 20-0GRP2 20-0GRP3 20-0GRP4 20-0GRP5 25-0GRP1 25-0GRP2
Lt = N(N-l)/2 Mes
(units/s) 395 594 471 468 636 490 463 459 499 533 514 546 685 808 707 827 787 1561 1468 1539 1425 2284 2422 2297
DELAY (s x 100) 2 78 4 54 3 95 4 00 3 89 2 64 2 39 3 77 3 87 3 03 3 91 2 99 5 96 4 90 3 50 3 92 5 47 6 44 711 8 26 5 29 7 67 7 61 8 09
CPU (s)
39 49 33 19 122 52 17 32 35 28 36 18 77 198 49 176 64 508 1140 595 823 1783 2046 2968
TABLE 1 Lt = N(N-l)j4 Mes
(units/s) 395 594 471 468 636 490 463 459 499 537 514 546 685 808 707 827 787 1561 1468 1539 1425 2270 2422 2 297
DELAY (s x 100)
2 78 4 54 3 95 4 01 3 91 2 64 2 40 3 77 3 87 3 06 3 91 2 99 5 96 4 90 3 50 3 92 5 47 6 44 711 8 26 5 29 8 08 7 61 8 09
CPU (s)
19 23 9 8 7 9 8 4 9 3 10 14 51 35 21 59 80 313 407 286 438 1454 1178 945
A - Mes
Pas (units/s)
395 594 471 468 636 494 463 459 499 533 501 538 690 808 707 820 787 1561 1468 1539 1425 2284 2422 2312
N(N-l)l DELAY (sx 100) 2 78 4 54 3 95 4 02 3 89 2 60 2 39 3 77 3 87 3 03 4 44 3 21 5 99 4 90 3 50 4 22 5 47 644 7 11 8 26 5 29 7 75 7 61 7 81
8 CPU
(s) 7 9 12 4 5 3 3 3 6 5 3 5 12 16 12 10 39 95 96 101 90 169 380 773
it can be easily seen that in almost ail cases the Lt = N(N—l)/4 schedule performs equally well, m terms of solutions obtamed, while the CPU time consumed is much less than half of the time of the Lt = N(N~ l)/2 schedule.
Obviously, "reasonably good" and CPU time-efficient solutions of the PAP are obtained using the Lt = N(N~ l)/8 schedule.
In tables 3 and 4 the message passing and queueing delay costs for two test problems with w varying from 0.2 to 1 are reported, The correspondmg number of processors used is also included. As expected, as w mcreases, the number of processors used and the amount of message passing decreases, while the amount of queueing delay mcreases.
QUEUING NETWORK IN TELECOMMUNICATIONS DISTRIBUTED SYSTEM 419
w — O ?W \J £,
DATA SET 12-3GRP1 12-3GRP2 12-3GRP3 12-3GRP4 12-3GRP5 12-1GRP1 12-1GRP2 12-0GRP1 12-0GRP2 12-0GRP3 12-0GRP4 12-0GRP5 15-0GRP1 15-0GRP2 15-0GRP3 15-0GRP4 15-OGRP5 20-OGRP1 20-0GRP2 20-0GRP3 20-0GRP4 20-0GRP5 25-0GRP1 25-0GRP2
Mes Pas (units/s)
431 656 509 511 713 569 565 523 576 619 599 612 771 853 762 849 840 1626 1515 1620 1500 2443 2490 2402
-N{N-\
DELAY (s x 100) 2 12 3 36 3 13 3 05 3 01 184 148 2 52 2 85 251 2 72 2 06 4 38 4 20 3 05 3 65 4 60 5 40 6 16 7 08 4 56 6 04 6 75 6 72
)/2 CPU
(s) 99 185 62 90 64 31 72 32 15 59 46 56 243 2 676 98 93 370 1496 4223 3 777 887 5 532 1841 3770
TABLE
4 = Mes Pas (units/s)
431 656 509 511 713 552 551 523 576 619 599 612 771 853 762 849 840 1625 1515 1620 1500 2443 2490 2402
2 N(N- 1)/
DELAY (sxlOO) 2 12 3 36 3 13 3 05 3 01 192 151 2 52 2 65 2 51 2 72 2 06 4 38 4 20 3 05 3 65 4 60 5 40 6 16 7 08 4 56 6 04 6 75 6 72
4 CPU
(s) 17 17 10 18 22 5 17 9 8 19 17 9 49 259 89 94 140 2324 182 270 173 653 1390 1066
Lt = Mes
Pas (units/s)
431 656 509 511 713 555 569 520 576 619 590 612 771 853 762 858 825 1625 1531 1620 1507 2457 2491 2417
N(N- 1)/
DELAY (s x 100) 2 12 3 36 3 13 3 05 3 01 190 148 2 59 2 65 2 51 2 76 2 06 4 38 4 20 3 05 3 65 4 65 5 40 6 14 7 08 4 56 6 08 6 75 6 72
8 CPU
(s) 6 7 8 5 7 4 7 3 5 5 4 5 20 23 14 13 13 100 55 108 37 119 160 237
TABLE 3 TABLE 4
w
0 2 0 4 0 6 0 8 1 0
4 = Mes Pas
(units/s) 565 540 529 463 429
N(N- l)/2 DELAY
(s x 100) 1 48 154 160 2 39 4 33
N°
Proc 9 7 7 4 4
w
0 2 0 4 0 6 0 8 10
Lt = N(N-\)/2 Mes Pas
(umts/s) 1620 1603 1582 1539 1491
DELAY (s x 100) 7 08 7 15 7 35 8 26 12 56
N°
Proc 16 15 14 13 10
DATA SET 12-1GRP2 DATA SET 20-0GRP3
6. CONCLUSION
In this paper the Process Allocation Problem is examined with the aim to minimize a combined objective function composée of queueing delay and communication costs and at the same time satisfy memory and occupancy constraints imposed on the network of identical processors. The construction of the queueing network, through the présentation of the congestion com- ponents in a télécommunications environment, and its solution, using Jack- son's model, are discussed. A simulated annealing algorithm was employed to tackle the problem. Computational results with a set of test problems, which have similar structure to a real world télécommunications problem, are reported indicating that the algorithm performs very well within reasona- ble amount of CPU time.
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